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June  2021, 10(2): 249-258. doi: 10.3934/eect.2020064

Decay rate of global solutions to three dimensional generalized MHD system

Yanxia Niu 1, , Yinxia Wang 2,, and  Qingnian Zhang 2,

1. 

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
2. 

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

* Corresponding author: Yinxia Wang

Received  September 2019 Published  June 2021 Early access  June 2020

Fund Project: The second author is supported in part by the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No. 20ZX002)

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space $ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $ follows by constructing time weighted energy inequality.

Keywords: generalized MHD system, decay rate of global solutions, analyticity.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35B40.
Citation: Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

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J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

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C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

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X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

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Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

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C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

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F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

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Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

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Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

show all references

References:
[1]

H. Bae, Existence and analyticity of Lei-Lin solution to Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar

[2]

J. Benameur, Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations, J. Math. Anal. Appl., 422 (2015), 424-434.  doi: 10.1016/j.jmaa.2014.08.039.  Google Scholar

[3]

J. Benameur and M. Bennaceur, Large time behavior of solutions to the 3D-NSE in spaces $\mathcal {X}^{\sigma}$, preprint, arXiv: 1901.09122v1. Google Scholar

[4]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875.  doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

C. He, X. Huang and Y. Wang, On some new global existence results for 3D magnetohydrodynamic equations, Nonlinearity, 27 (2014), 343-352.  doi: 10.1088/0951-7715/27/2/343.  Google Scholar

[8]

X. Jia and Y. Zhou, On regularity criteria for the 3D incompressible MHD equations involving one velocity component, J. Math. Fluid Mech., 18 (2016), 187-206.  doi: 10.1007/s00021-015-0246-1.  Google Scholar

[9]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[10]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[11]

F. G. Liu and Y.-Z. Wang, Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew Math. Phys., 70 (2019), no. 3, Paper No. 69, 12 pp. doi: 10.1007/s00033-019-1113-3.  Google Scholar

[12]

Y. Lin, H. Zhang and Y. Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations, 261 (2016), 102-112.  doi: 10.1016/j.jde.2016.03.002.  Google Scholar

[13]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  doi: 10.1002/mma.1026.  Google Scholar

[14]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

W. Wang, T. Qin and Q. Bie, Global well-posedness and analyticity results to 3D generalized magnetohydrodynamics equations, Appl. Math. Lett., 59 (2016), 65-70.  doi: 10.1016/j.aml.2016.03.009.  Google Scholar

[17]

Y.-Z. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[18]

Y.-Z. Wang and P. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods. Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

Y. Wang, Asymptotic decay of solutions to 3D MHD equations, Nonlinear Anal., 132 (2016), 115-125.  doi: 10.1016/j.na.2015.11.002.  Google Scholar

[20]

Y. Xiao, B. Yuan and Q. Zhang, Temporal decay estimate of solutions to 3D generalized magnetohydrodynamics system, Appl. Math. Lett., 98 (2019), 108-113.  doi: 10.1016/j.aml.2019.06.003.  Google Scholar

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[22]

Z. Ye and X. P. Zhao, Global well-posedness of the generalized magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (2018), 1-26.  doi: 10.1007/s00033-018-1021-y.  Google Scholar

[23]

Z. Zhang and Z. Y. Yin, Global well-posedness for the generalized Navier-Stokes system, preprint, 2013, arXiv: 1306.3735v1. Google Scholar

[24]

Z. J. Zhang, Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.  doi: 10.1080/00036811.2016.1207245.  Google Scholar

[25]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar

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